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Mixing, Transport and Turbulence Modulation in Solid Suspensions: Study and Modelling

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Mixing, Transport and Turbulence Modulation in Solid Suspensions: Study and Modelling Mixing, transport and turbulence modulation in solid suspensions : study and modelling François Laenen To cite this version: François Laenen. Mixing, transport and turbulence modulation in solid suspensions : study and modelling. Other [cond-mat.other]. Université Côte d’Azur, 2017. English. NNT : 2017AZUR4010. tel-01534441 HAL Id: tel-01534441 https://tel.archives-ouvertes.fr/tel-01534441 Submitted on 7 Jun 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Université Côte d’Azur École doctorale de Sciences Fondamentales et Appliquées Unité de recherche : Laboratoire Joseph-Louis Lagrange – UMR 7293 Thèse de doctorat Présentée en vue de l’obtention du grade de Docteur en Sciences de Université Côte d’Azur Discipline : Physique Présentée et soutenue par François Laenen Mixing, transport and turbulence modulation in solid suspensions Study and modelling Dirigée par Jérémie Bec, Directeur de Recherche CNRS, Observatoire de la Côte d’Azur et codirigée par Giorgio Krstulovic, Chargé de Recherche CNRS, Observatoire de la Côte d’Azur Soutenue le 24 février 2017 Devant le jury composé de : Sergio Chibbaro Maitre de Conférence HDR, Université Pierre et Marie Curie Rapporteur Romain Volk Maitre de Conférence HDR, ENS Lyon Rapporteur Guido Boffetta Professeur, Université de Turin, Italie Examinatrice Aurore Naso CR CNRS, Ecole centrale Lyon Examinatrice Mikhael Gorokhovski Professeur, Ecole centrale Lyon Examinateur Emmanuel Villermaux Professeur, Aix-Marseille Université Examinateur Jérémie Bec DR CNRS, Université Côte d’Azur Directeur Giorgio Krstulovic CR CNRS, Université Côte d’Azur Co-directeur ii Summary The transport of particles by turbulent flows is ubiquitous in nature and industry. It occurs in planet formation, plankton dynamics and combustion in engines. For the dispersion of atmospheric pollutants, traditional predictive models based on eddy diffusivity cannot accurately reproduce high concentration fluctuations, which are of primal importance for ecological and health issues. The first part of this thesis relates to the dispersion by turbulence of tracers continuously emitted from a point source. Mass fluctuations are characterized as a function of the distance from the source and of the observation scale. The combination of various physical mixing processes limits the use of fractal geometric tools. An alternative approach is proposed, allowing to interpret mass fluctuations in terms of the various regimes of pair separation in turbulent flows. The second part concerns particles with a finite and possibly large inertia, whose dispersion in velocity requires developing efficient modelling techniques. A novel numerical method is proposed to express inertial particles distribution in the position-velocity phase space. Its convergence is validated by comparison to Lagrangian measurements. This method is then used to describe the modulation of two-dimensional turbulence by large-Stokes-number heavy particles. At high inertia, the effect is found to be analogous to an effective large-scale friction. At small Stokes numbers, kinetic energy spectrum and nonlinear transfers are shown to be modified in a non trivial way which relates to the development of instabilities at vortices boundaries. R´esum´e Le transport de particules par des ´ecoulements turbulents est un ph´enom`ene pr´esent dans de nom- breux ´ecoulements naturels et industriels, tels que la dispersion de polluants dans l’atmosph`ereou du phytoplancton et plastiques dans et `ala surface des oc´eans. Les mod`elespr´edictifs classiques ne peuvent pr´evoir avec pr´ecision la formation de larges fluctuations de concentrations. La premi`erepartie de cette th`ese concerne une ´etude de la dispersion turbulente de traceurs ´emis `apartir d’une source ponctuelle et continue. Les fluctuations spatiales de masse sont d´etermin´ees en fonction de la distance `ala source et `al’´echelle d’observation. La combinaison de plusieurs ph´enom`enes physiques `al’origine du m´elange limite la validit´e d’une caract´erisationde g´eom´etrie fractale. Une approche alternative est propos´ee, permettant d’interpr´eter les fluctutations massiques en terme des diff´erents r´egimes de s´eparation de pair dans des ´ecoulements turbulents. La seconde partie concerne des particules ayant une inertie finie, dont la dispersion dans l’espace des vitesses requiert de d´evelopper des techniques de mod´elisationadapt´ees. Une m´eth- ode num´erique originale est propos´ee pour exprimer la distribution des particles dans l’espace position-vitesse. Cette m´ethode est ensuite utilis´eepour d´ecrirela modulation de la turbulence bi-dimensionnelle par des particules inertielles. A grand nombres de Stokes, l’effet montr´eest ana- logue `acelui d’une friction effective `agrande ´echelle. Aux petits Stokes, le spectre de l’´energie cin´etique du fluide et les transferts non-lin´eaires sont modif´ees d’une mani`ere non triviale. Contents 1 Introduction and context 1 2 Definitions and concepts 11 2.1 Navier–Stokes equations ............................. 11 2.1.1 Structure functions and intermittency ................. 12 2.2 Turbulence in two dimensions .......................... 15 2.2.1 Two dimensional Navier–Stokes equations ............... 15 2.2.2 The double cascade framework ..................... 17 2.2.3 Energy and enstrophy budgets ..................... 21 2.3 Relative dispersion rates of Lagrangian trajectories .............. 22 2.3.1 Lyapunov exponents ........................... 22 2.3.2 Separation rates ............................. 23 I Turbulent dispersion and mixing 27 3 Tracers dispersion in two dimensional turbulence 29 3.1 Introduction .................................... 29 3.1.1 Diffusion at long times .......................... 33 3.1.2 Continuous source ............................ 36 3.2 middling version ................................. 36 3.2.1 Fluid phase integration ......................... 36 3.2.2 Injection mechanism ........................... 38 3.2.3 Removal mechanism ........................... 39 3.3 Results ....................................... 39 iii iv CONTENTS 3.3.1 One point dispersion ........................... 39 3.3.2 Two-point correlation .......................... 43 3.3.3 Phenomenological description ...................... 49 3.4 Brief conclusion .................................. 54 II Inertial particle-laden flows 55 4 A lattice method for the numerical modelling of inertial particles 57 4.1 Inertial particles dynamics ............................ 57 4.1.1 Individual particles ............................ 57 4.2 The modelling of dispersed multiphase flows .................. 61 4.2.1 From microscopic description to macroscopic quantities ....... 62 4.3 Description of the method ............................ 67 4.4 Application to a one-dimensional random flow ................. 70 4.4.1 Particle dynamics for d =1 ....................... 70 4.4.2 Lattice-particle simulations ....................... 74 4.5 Application to incompressible two-dimensional flows ............. 79 4.5.1 Cellular flow ............................... 79 4.5.2 Heavy particles in 2D turbulence .................... 81 4.6 Conclusions .................................... 86 5 Turbulence modulation by small heavy particles 89 6 Conclusions and perspectives 113 6.1 Turbulent transport of particles emitted from a point source ......... 113 6.2 Modelisation of small inertial particles ..................... 115 6.3 Turbulence modulation by small heavy particles ................ 116 Appendices 119 A Software details 121 A.1 GPU2DSOLVER ................................. 121 A.1.1 Numerical scheme ............................ 121 A.1.2 Forcing .................................. 123 A.1.3 Parallelisation .............................. 124 A.2 LAGSRC2D .................................... 126 A.2.1 Numerical implementation ........................ 126 A.2.2 Injection rate ............................... 127 A.3 LOCA: Lattice model for heavy particles .................... 128 A.3.1 Finite volume fluxes ........................... 128 A.3.2 Dynamic grid resizing (DGR) ..................... 129 CONTENTS v A.3.3 Parallelisation .............................. 131 B SoAx: a convenient and efficient C++ library to handle simulation of heterogeneous particles in parallel architectures 133 Acknowledgments .................................... 159 vi CONTENTS Nomenclature V Tracer velocity. Vp Inertial particle velocity. X Tracer position. Xp Inertial particle position. xS Position of the injecting source. δru Longitudinal velocity increment between two particles separated by a distance r. ✏ Total energy dissipation rate. ✏↵ Energy dissipation rate due to large-scale friction. ✏⌫ Energy dissipation rate due to viscous friction. Molecular diffusion coefficient. T Turbulent diffusion coefficient. V Diffusivity in the velocity space. φm Massic ratio between the solid and the fluid phase. −1 φS Source injection rate. Units are T . φv Volumic fraction of the particles. Fluid stream function. vii viii CONTENTS ⇢(x) Fluid density field. ⇢p(x) Particle density field. ⇢p(x)=np(x)Mp
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